I use J&J baby shampoo for my eye makeup. It works better than anything else I've tried and it never stings. Also if you're prone to styes it supposedly helps keep that whole thing super clean and free of blockage. If I'm travelling I use E.L.F. eye makeup mini rounds. They always dry out within a few days, but they work fine to just re-wet.
I use a Clarisonic Mia with Burt's Bees Soap Bark and Chamomile cleansing cream all over to deal with face makeup and any eye makeup that got smeared/dripped. Then if there's any liner or anything left over, I take a tiny bit of face cream on my finger or a q-tip and it comes right off.
If I'm somehow caught spending the night somewhere I can't wash my face and don't have anything with me, I wash my hands and slather them in lotion, then rub my eyes until I look like a greasy raccoon, then I wipe it off with a towel. It all comes right off with no tugging.
First I use the Simple Eye Makeup Remover from Target, then wipes. I'm not too picky...sometimes Ponds, sometimes Target brand, sometimes Epielle (the ones from Big Lots). I'm using Epielle ones right now.
For some reason, all makeup removers and wipes break me out, so I wash my face first with Pevonia facial cleanser. Then I use my Clarisonic Mia and then I use Pevonia toner to make sure that I got every last bit of makeup off my face. This works well for me since I don't wear a ton of eye makeup and the makeup that I do wear is not waterproof.
I use Caudalie Gentle Cleanser which is gentle enough to use on the eyes. Dissolves all of my makeup. My Clarisonic Mia helps take most of it off and I can wipe the last little bit with a washcloth when I dry my face. I follow with Caudalie Toning Lotion but generally all the makeup has been removed by then.
I hate using an additional eye makeup remover but most of the cleanser/makeup remover combos have either irritated my skin or my eyes. The Caudalie is the first I've tried that is gentle enough for my sensitive skin and eyes but that actually still works.
I used to be an avid user of Pond's Cold Cream to remove eyemakeup, but just recently I picked up theBalm Balms Away Eye Makeup Breakup. This stuff removes my eye makeup in one wipe. Leave's the skin in my periocular area super moisturized afterwards. For face makeup, I use good ol' Cetaphil facial wash and my Clarisonic.
I use wet wipes for the first removal and then wash my face followed by moisturiser. It's a mini pamper session. For those days when I'm wearing the whole kit and kaboddle I sometime also use a lovely cleansing oil which I can't remember the name of right now.